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Once again on thin-shell wormholes in scalar-tensor gravity
arXiv:0903.5173v3 [gr-qc] 2 Jul 2009
Kirill A. Bronnikov a,1 and Alexei A. Starobinsky b,c,2 a
Center of Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya St., Moscow 117361, Russia; Institute of Gravitation and Cosmology, PFUR, 6 Miklukho-Maklaya St., Moscow 117198, Russia
b
Landau Institute for Theoretical Physics of RAS, Moscow 119334, Russia
c
RESCEU, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan
It is proved that all thin-shell wormholes built from two identical regions of vacuum static, spherically symmetric space-times have a negative shell surface energy density in any scalar-tensor theory of gravity with a non-ghost massless scalar field and a non-ghost graviton.
It has been recently proved in a general form [1] that no wormholes can be formed in any scalartensor theory (STT) of gravity in which the non-minimal coupling function f (Φ) is everywhere positive and the scalar field Φ itself is not a ghost if matter sources of gravity respect the Null Energy Condition (NEC). In order to construct any viable and stable wormhole solution, attempts have been recently made to find such a solution in STT of gravity using thin shells instead of extended matter sources, and it has been claimed that at least in some cases (namely, in the Brans-Dicke STT for a particular range of values of the coupling constant ω ) the shell at the wormhole throat may satisfy the weak and null energy conditions [2]. In this short note we will show explicitly that, in any STT with a massless non-ghost scalar field, in all thin-shell wormholes built from two identical regions of vacuum static, spherically symmetric space-times, the shell has negative surface energy density (note that we here do not consider numerous toy wormhole models which do not represent solutions of any initially fixed equations of some metric theory of gravity). Let us begin with presenting the nonzero components of the Einstein tensor Gνµ = Rµν − 21 δµν R for a general static, spherically symmetric space-time with the metric3 ds2 = e2γ(u) dt2 − e2α(u) du2 − e2β(u) dΩ2 ,
(1)
where dΩ2 = dθ2 + sin2 θdζ 2 is the metric on a unit sphere and u is an arbitrarily chosen radial coordinate. We have (the prime denotes d/du) G00 = e−2α (2β ′′ + 3β ′2 − 2α′β ′ ) − e−2β , G11 = e−2α (β ′2 + 2β ′ γ ′ ] − e−2β , G22 = G33 = e−2α [γ ′′ + γ ′2 + β ′′ + β ′2 + β ′ γ ′ − α′ (β ′ + γ ′ )]. 1
(2)
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[email protected] 3 Our conventions are: the metric signature (+ − − −); the curvature tensor Rσ µρν = ∂ν Γσµρ − . . . , Rµν = Rσ µσν , so that the Ricci scalar R > 0 for de Sitter space-time and the matter-dominated cosmological epoch; the system of units 8πG = c = 1 . 2
2 Now, consider Fisher’s well-known solution [3] to the Einstein — massless scalar equations which can be written in the form ds2 = P a dt2 − P −a dr 2 − P 1−a r 2 dΩ2 , C ψ = − ln P (r), 2k C2 2k a2 = 1 − 2 , P = P (r) := 1 − , 2k r
(3) (4) (5)
where ψ is the scalar field, a, C and k are integration constants related as given in (5). The metric (3) is asymptotically flat, with the Schwarzschild mass m equal to ak ; note that a2 ≤ 1, and in case a = 1, C = 0 the Schwarzschild solution is restored. The solution (3)–(5) with C 6= 0 has a naked singularity at r = 2k , situated at the centre of symmetry since there g22 = 0 (the coordinate spheres shrink to a point). However, following [2] and some other papers, one can easily obtain a traversable wormhole geometry using the cut-and-paste trick [4]: one takes two copies of the region r ≥ r0 > 2k of the space-time (3) and identifies the spheres r = r0 in them. This procedure is formally described as putting r = r0 + |u|
(6)
in (3)–(5), where u is a new radial coordinate, and assigning u ≤ 0 to one copy of the region r ≥ r0 and u ≥ 0 to the other. Then the derivatives of gµν are discontinuous at u = 0, and this may be ascribed to appearance of a thin shell with certain energy density and tension. The latter can be found by substituting the solution (3)–(5) rewritten in terms of u according to (6) into the Einstein equations Gνµ = −Sµν (ψ) − Tµν ,
(7)
where Sµν (ψ) = ψ ,ν ψ,µ − 21 δµν (∂ψ)2 is the stress-energy tensor (SET) of the field ψ while Tµν is the shell SET proportional to Dirac’s delta function, δ(u). Outside the shell, Eqs. (7) are manifestly satisfied by our solution, and the task is to find Tµν = δ(u) diag(σ, 0, −p⊥ , −p⊥ ),
(8)
where σ is the surface density and p⊥ the surface pressure; the radial pressure should evidently vanish because the shell is perpendicular to the radial direction. Nonzero contributions to σ and p⊥ appear only due to d2 r/du2 = 2δ(u). Therefore, to find them, in the expressions (2) it is sufficient to take into account only terms with second-order derivatives since all other terms are finite at u = 0 (to be denoted by the symbol “fin”). In particular, Tµν (ψ) = fin and G11 = fin because they contain only first-order derivatives. A straightforward calculation then gives 4δ(u) a−1 P [r0 − (1 + a)k] + fin, r02 0 2δ(u) T22 = T33 = − 2 P0a−1 (r0 − k) + fin, r0 T00 = −
T11 = fin, (9)
where P0 = P (r0 ) = 1 − 2k/r0 . Since r0 > 2k and |a| < 1, the surface density σ is negative. We have thus calculated the shell characteristics directly from the Einstein equations without invoking the well-known Israel formalism for thin shells; our method is similar to what was done in [5] in the case of conical singularities (infinitely thin cosmic strings). The Israel formalism would
3 be quite necessary if we wished to study the shell dynamics, but in a purely static description our direct method is simpler and more transparent. Now, we can recall that the Einstein-scalar equations, whose solution is given by (3)–(5), can be regarded as the Einstein-frame equations of an arbitrary STT with the Jordan-frame Lagrangian (written in the Brans-Dicke parametrization for a space-time manifold MJ with the metric gµν ) 1 ω(φ) µν LJ = φR + g φ,µ φ,ν − 2U(φ) + Lm , (10) 2 φ where R is the Ricci scalar, Lm is the Lagrangian of nongravitational matter, ω(φ) and U(φ) are arbitrary functions. In the general case, transition to the Einstein frame, defined as a manifold ME with the metric g µν = |φ|gµν ,
(11)
results in the Lagrangian h i 1 LE = (sign φ) R + [sign(ω + 3/2)]g µν ψµ ψ,ν − V (ψ) + LmE , 2
(12)
where bars mark quantities obtained from or with g µν , indices are raised and lowered with g µν and p |ω + 3/2| dψ = , V (ψ) = φ−2 U(φ). LmE = φ−2 Lm . (13) dφ |φ| The above relations describing a thin-shell wormhole represent a solution to the field equations corresponding to the Lagrangian (12), where the matter Lagrangian LmE leads to the SET (9), the potential V (ψ) ≡ 0 (the scalar field is massless), and the following sign conditions hold: φ > 0 (which means that the graviton is not a ghost) and ω + 3/2 > 0 (the ψ field has a normal sign of kinetic energy, and, equivalently, the φ field is not a ghost). Suppose a particular STT is chosen, satisfying the above sign conditions, with a certain function ω(φ) and U(φ) ≡ 0. Then, according to (11) and (13), the metric i 1h a 2 2 −a 2 1−a 2 2 dsJ = P dt − P dr − P r dΩ , (14) φ with the notations (5) and the scalar field φ related to ψ as given in (13), satisfy the field equations due to (10) with Lm = 0 and represent a scalar-vacuum solution of the theory (10), discussed in a general form in [6]. Moreover, the result of the cut-and-paste procedure described above, applied to (14) with the substitution (6), is, in general, a wormhole configuration with a thin shell at the throat r = r0 whose SET Tµν has the form Tµν = φ2 Tµν with Tµν given by (9). This means that such wormholes are supported by thin shells with negative energy density in any STT. This is a direct consequence of the relations between the Jordan and Einstein frames. For better clarity, let us confirm this conclusion by a direct calculation. The metric field equations following from (10) read (Gνµ + ∇µ ∇ν − δµν 2)φ +
ω(φ) ν S (φ) + δµν U(φ) = −Tµν , φ µ
(15)
where 2 = ∇α ∇α is the d’Alembert operator; Sµν (φ) = φ,ν φ,µ − 21 δµν (∂φ)2 ; in the case under √ √ consideration, U ≡ 0; and Tµν = (2/ −g)(δLm −g/δg µν ) is the SET corresponding to Lm (all operations being performed with the metric gµν involved in (10)).
4 Substituting the metric (14) with (6) into (15), we are again interested only in terms proportional to δ(u). Therefore, we can write G00 = 2 e−2α β ′′ + fin, G11 = fin, G22 = e−2α (β ′′ + γ ′′ ) + fin, 2δ(u) 1 φ′′ ′′ β = 2 [r − k(1 + a)] − + fin, r P 2 φ 2ak δ(u) 1 φ′′ − + fin, γ ′′ = r2P 2 φ (∇0 ∇0 − 2)φ = (∇2 ∇2 − 2)φ + fin = e−2α φ′′ + fin, (∇1 ∇1 − 2)φ = fin,
Sµν (φ) = fin,
(16)
where the metric (14) is identified with (1), so that, e.g., e−2α = φP a ; as before, P = P (r) and the prime means d/du. As a result, we obtain 4φ2 δ(u) a−1 P0 [r0 − (1 + a)k] + fin, r02 2φ2 δ(u) a−1 T22 = T33 = − P0 (r0 − k) + fin, r02 T00 = −
T11 = fin, (17)
so that Tµν = φ2 Tµν , as required. It is of interest that this calculation did not involve the relation between φ and ψ from (13); it was only assumed that φ(r) is finite and smooth, so that only φ′′ could lead to expressions with δ(u). Let us use the Brans-Dicke theory, with ω = const > −3/2, as a particular example. In this case, as follows from (13), we can put p C , (18) φ = exp[ψ/ ω + 3/2] = P ξ , ξ := − p 2k ω + 3/2 where ξ is an effective scalar charge. The relation between constants in (5) is converted to a2 + (2ω + 3)ξ 2 = 1.
(19)
Then in (14) and (16) we have e2γ = P a−ξ ,
e−2α = P a+ξ ,
e2β = rP 1−a−ξ ,
and a direct calculation confirms the result (17).4 Inclusion of radial electromagnetic fields would make the solutions slightly more complex and diverse [6] but the main result would be the same: in a ghost-free STT, thin-shell wormholes can be obtained with negative surface shell densities only, violating the weak energy condition. For comparison, it can be recalled that if we admit ω < −3/2 in the Lagrangian (10), which makes the φ field a ghost, then wormholes exist already among vacuum and electrovacuum solutions of such theories [6] without any shells. It has been shown, however, that at least some vacuum solutions of this class are unstable [7, 8] (see also [9]). AS acknowledges RESCEU hospitality as a visiting professor. He was also partially supported by the grant RFBR 08-02-00923 and by the Scientific Programme “Astronomy” of the Russian Academy of Sciences. KB acknowledges hospitality at Institut de Ci`encies de l’Espai (Barcelona), where part of the work was done, and partial support from NPK MU grant of Peoples’ Friendship University of Russia. 4
Though we are using the same coordinate r in the same Brans-Dicke scalar-vacuum solution as in [2], our notations for the constants are different: their constants η, A, B are equal to our k, a − ξ, −(a + ξ), respectively.
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References [1] K. A. Bronnikov and A. A. Starobinsky, JETP Lett. 85, 1 (2007) (arXiv:gr-qc/0612032). [2] E. F. Eiroa, M. G. Richarte and C. Simeone, Phys. Lett. A 373, 1 (2008) (arXiv:0809.1623). [3] I. Z. Fisher, Zh. Eksp. Teor. Fiz. 18, 636 (1948) (arXiv:gr-qc/9911008). [4] M. Visser, Phys. Rev. D 39, 3182 (1989). [5] D. D. Sokolov and A. A. Starobinsky, Sov. Phys. - Doklady 22, 312 (1977). [6] K. A. Bronnikov, Acta Phys. Pol. B4, 251 (1973). [7] J. A. Gonzalez, F. S. Guzman and O. Sarbach, Class. Quantum Grav. 26, 015010 (2009) (arXiv:0806.0608). [8] J. A. Gonzalez, F. S. Guzman and O. Sarbach, Class. Quantum Grav. 26, 015011 (2009) (arXiv:0806.1370). [9] A. Doroshkevich, J. Hansen, I. Novikov and A. Shatskiy, arXiv:0812.0702.
